$ A = \left[\begin{array}{rrr}5 & 3 & -1 \\ 1 & 4 & 5\end{array}\right]$ $ C = \left[\begin{array}{rr}3 & 1 \\ -1 & 2 \\ 2 & 2\end{array}\right]$ What is $ A C$ ?
Solution: Because $ A$ has dimensions $(2\times3)$ and $ C$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ A C = \left[\begin{array}{rrr}{5} & {3} & {-1} \\ {1} & {4} & {5}\end{array}\right] \left[\begin{array}{rr}{3} & \color{#DF0030}{1} \\ {-1} & \color{#DF0030}{2} \\ {2} & \color{#DF0030}{2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rr}{5}\cdot{3}+{3}\cdot{-1}+{-1}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{3}+{3}\cdot{-1}+{-1}\cdot{2} & ? \\ {1}\cdot{3}+{4}\cdot{-1}+{5}\cdot{2} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{5}\cdot{3}+{3}\cdot{-1}+{-1}\cdot{2} & {5}\cdot\color{#DF0030}{1}+{3}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{2} \\ {1}\cdot{3}+{4}\cdot{-1}+{5}\cdot{2} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{5}\cdot{3}+{3}\cdot{-1}+{-1}\cdot{2} & {5}\cdot\color{#DF0030}{1}+{3}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{2} \\ {1}\cdot{3}+{4}\cdot{-1}+{5}\cdot{2} & {1}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}10 & 9 \\ 9 & 19\end{array}\right] $